Countably compact spaces with \(n\)-in-countable weak bases are metrizable
type: Item
instance of: scholarly article
title: Countably compact spaces with \(n\)-in-countable weak bases are metrizable
zbMATH Open document ID: 1305.54038
author: Zuoming Yu
published in: Topology and its Applications
publication date: 2014-08-26T00:00:00Z
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D70
zbMATH DE Number: 6334514
zbMATH Keywords: weak base
zbMATH Keywords: countably compact space
zbMATH Keywords: monotonically monolithic space
zbMATH Keywords: \(n\)-in-countable collection
zbMATH Keywords: metrization theorem
zbMATH Keywords: D-space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: j.topol.2014.07.008
OpenAlex ID: W2008898953
cites work: MAPPINGS AND SPACES
cites work: Q3715852
cites work: Sharp bases and weakly uniform bases versus point-countable bases
cites work: A broader context for monotonically monolithic spaces
cites work: Q3894636
cites work: Q4076488
cites work: Monotonically monolithic spaces, Corson compacts, and \(D\)-spaces
cites work: \(D\)-spaces and thick covers
cites work: Q4080409
cites work: Q5617506
cites work: Classes defined by stars and neighbourhood assignments
cites work: Q3083848
cites work: Monolithic spaces and \(D\)-spaces revisited
cites work: The Collins-Roscoe property and its applications in the theory of function spaces
cites work: Lifting the Collins-Roscoe property by condensations
cites work: On spaces with \(k\)-in-countable bases or weak bases
cites work: A note on the extent of two subclasses of star countable spaces
DOI: 10.1016/J.TOPOL.2014.07.008
review text: Following \textit{A. V. Arhangels'skij} [Russ. Math. Surv. 21, No.4, 115--162 (1966); translation from Usp. Mat. Nauk 21, No .4(130), 133--184 (1966; Zbl 0171.43603)], a collection \({\mathcal B}= \bigcup\{{\mathcal B}_x\mid x\in X\}\) of subsets of a topological space \(X\) is called a weak base of \(X\) if, for each \(x\in X\), \({\mathcal B}_x\) is closed under finite intersections and \(x\in\bigcap{\mathcal B}_x\), and a subset \(U\) of \(X\) is open if and only if for each \(x\in U\), there is a \(B\in{\mathcal B}_x\) such that \(x\in B\subset U\).NEWLINENEWLINE In a recent paper, the author and \textit{Z. Yun} proved some interesting results concerning Tychonoff spaces with an \(n\)-in-countable (weak) base, where, for every positive integer \(n\), a collection \({\mathcal B}\) of subsets of a set \(X\) is called \(n\)-in-countable if every \(A\subset X\) with \(|A|=n\) is contained in only finitely many elements of \({\mathcal B}\) [Topology Appl. 159, No. 17, 3545--3549 (2012; Zbl 1256.54051)]. Some of the problems that were left open in this paper have been solved in the reviewed paper.NEWLINENEWLINE The main result is mentioned in the title. It answers a question raised by \textit{H. R. Bennett} and \textit{H. W. Martin} [Port. Math. 36, 247--250 (1977; Zbl 0448.54032)]. Its interesting proof uses the non-trivial fact, first proven by \textit{V. V. Tkachuk} [Topology Appl. 156, 840--846 (2009; Zbl 1165.54009)], that every monotonically monolithic Tychonoff space is a D-space. Every Tychonoff space with an \(n\)-in-countable weak base for some \(n\) is weakly monotonically monolithic, and every Tychonoff space with an \(n\)-in-countable base for some \(n\) is monotonically monolithic. There are several more interesting results concerning Tychonoff spaces with an \(n\)-in-countable base, which are omitted here.NEWLINENEWLINE Finally, two questions of \textit{V. V. Tkachuk} [Topology Appl. 159, No. 6, 1529--1535 (2012; Zbl 1245.54022)] are answered in the neg…
description: scientific article; zbMATH DE number 6334514